You mean you want an expression that is human-readable even when the order is a symbol? Here is an attempt:

poly[vars_List, a_, order_Integer /; order >= 0] :=

 Module[{n = Length@vars, idx, z},
  idx = Select[Tuples[Range[0, order], n], Total[#] <= order &]; 
  Plus @@ 
   Table[Subscript[a, Sequence @@ r] Times @@ (vars^r), {r, idx}]]; 
poly[vars_List /; Length[vars] > 2, a_, order_] :=

 Inactive[Sum][
  Inactive[Times][
   Subscript[a, Subscript["i", 1], \[Ellipsis], 
    Subscript["i", Length[vars]]], vars[[1]]^
   Subscript["i", 1], \[Ellipsis], vars[[-1]]^
   Subscript["i", Length[vars]]], 
  0 <= Inactive[Plus][Subscript["i", 1], \[Ellipsis], 
    Subscript["i", Length[vars]]] <= order]

Try poly[{x, y, z, t}, a, n]

Hi Vinícius

One issue is that N is a built-in symbol. Try

ClearAll@n
poly[{x, y}, a, n]

Range[0, n] is unbounded. Is there a reason why you cannot use Sum to generate the polynomial?

Hi Mr. Schachner, I hope you are doing well.

Thank you for your insightful comment. Indeed, I have also tried to compute $P(x,y)$ using Sum. However, it did not work nicely in the general case. Nevertheless, your code has clarified all of my doubts.

Thanks once more, and God bless you.